Canonical heights for abelian group actions of maximal dynamical rank

Abstract

Let $X$ be a smooth projective variety of dimension $n\ge 2$ and $G\cong {\mathbf{Z}}^{n-1}$ a free abelian group of automorphisms of $X$ over $\overline{\mathbf{Q}}$. Suppose that $G$ is of positive entropy. We construct a canonical height function ${\hat{h}}_G$ associated with $G$, corresponding to a nef and big $\mathbf{R}$-divisor, satisfying the Northcott property. By characterizing its null locus, we prove the Kawaguchi--Silverman conjecture for each element of $G$. As another application, we determine the height counting function for non-periodic points.

27 pages, comments welcome!

References in corpus (6)

• Invariant subvarieties with small dynamical degree
• Eigenvalues and dynamical degrees of self-maps on abelian varieties
• Existence of the equivariant minimal model program for compact Kähler threefolds with the action of an abelian group of maximal rank
• Structures theorems and applications of non-isomorphic surjective endomorphisms of smooth projective threefolds
• Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics
• Kawaguchi-Silverman conjecture for int-amplified endomorphism